We consider generalized linear transient convection-diffusion problems for
differential forms on bounded domains in $\mathbb{R}^{n}$. These involve Lie
derivatives with respect to a prescribed smooth vector field. We construct both
new Eulerian and semi-Lagrangian approaches to the discretization of the Lie
derivatives in the context of a Galerkin approximation based on discrete
differential forms. Details of implementation are discussed as well as an
application to the discretization of eddy current equations in moving media.